6.2 The outer problem
6.2.1 Analytical studies
Preliminaries
We start by summarising some properties of the function uBS defined in (3.10), which will
be useful in the following. For a proof we refer to [22]. Lemma 6.2.1
Let the parameters P, K > 0 and r ≥ 0 be fixed. Then the function uBS(P, K, r, t, s) is
nonnegative and continuous for (t, s) ∈ [0, ∞] × [0, ∞). Moreover, for t ≥ 0 and s > 0 the following properties hold.
1. The function uBS is continuously differentiable with respect to s and it holds
∂uBS(P, K, r, t, s)
∂s = φ(d1) P 1
2√s > 0 (6.18) with d1 defined by (3.11) and the distribution function φ(z) = √12πe−
z2
2 of standard
normal distribution.
2. The function uBS is continuously differentiable with respect to t, where we have
∂uBS(P, K, r, t, s)
∂t = r K e
−r tΦ(d
2)≥ 0 . (6.19)
with d2 defined by (3.11).
3. Furthermore, we find the following limit conditions
lim s→0 ∂uBS(P, K, r, t, s) ∂s = ∞ if P = K e−r t 0, else (6.20) as well as lim s→∞uBS(P, K, r, t, s) = P . (6.21)
Clearly, the properties of the function uBS can also be formulated in terms of the generating
function f of the Nemytskii operator N. Let us formulate some important implications of Lemma 6.2.1
Corollary 6.2.2
For fixed parameters P0, K > 0 and r ≥ 0 the function f : I × [0, ∞) → [0, ∞) defined by
f (t, s) = uBS(P0, K, r, t, s) (t, s)∈ I × [0, ∞) (6.22)
6.2. THE OUTER PROBLEM 97
1. f (t, 0) = max(P0− Ke−rt, 0) and lims→∞f (t, s) = P0
2. The functions f (t, s) is strictly monotonically increasing with respect to s and mono- tonically non-decreasing with respect to t.
Especially, the function f possesses an extension ˜f : I× R → R that satisfies the condition (A) and (B) of Assumption 5.3.1 and is equal to f on the set I× [0, ∞).
Proof: Properties 1 and 2 follow immediately from the definition of f and Lemma 6.2.1. The function ˜f can be chosen as
˜ f (t, s) = f (t, s) s≥ 0 f (t, 0) + s s < 0
Clearly, ˜f (t, s) is continuous and strictly monotonically increasing with respect to s, which proves the condition (A) of Assumption 5.3.1.
Furthermore, the continuity of ˜f implies that ˜f is also a Carathéodory function. Besides, for every t∈ I we have
0≤ ˜f (t, s)≤ P0 whenever s≥ 0
and
s ≤ ˜f (t, s)≤ max(P0− Ke−rt, 0)≤ P0 whenever s < 0
Hence, (5.8) is satisfied with a = P0 and b = 1. Hence also the condition (B) of Assump-
tion 5.3.1 is fulfilled.
The L2-setting
In the following Subsubsection we are going to analyse the situation (b). Let P0, K > 0 and
r ≥ 0 be fixed and f : I × [0, ∞) → [0, ∞) be defined by (6.22). We start by proving that in this setting the inverse problem is ill-posed. Indeed, we have a very similar situation as in Example 5.3.9. Here it holds 0≤ f(t, s) ≤ P0 for all (t, s)∈ I × [0, ∞). We can therefore
consider the functions Sn(t) := t r≤ T − 1 n T1+ q t− T + 1 nn3/2 t≥ T − 1 n t ∈ I
and S0(t) := t t∈ I. Then we have
kSn− S0k2L2(I) = Z T T −1/n t− T + 1 n n3dt = n 2 and kN(Sn)− N(S0)k2L2(I2) ≤ Z T T −1/n P02dt≤ 1 nP0 2.
Hence,
N(Sn)→ N(S0) but Sn9S0, (6.23)
which proves that the inverse of N : N(D+)⊂ L2(I)→ L2(I) is not continuous.
Next we consider two regularization methods which are based on a restriction of the set of admissible solutions. The main point is to find a set ˜D such that N restricted to the set ˜D has a continuous inverse and the true function S† is an element of this set S† ∈ ˜D. Then we
can choose the approximate solution Sδ as minimizer of the extremal problem
kN( ˜S)− uδkL2(I) → min, subject to ˜S ∈ ˜D . (6.24)
In [23, Section 4] it has been concluded from the well known result formulated in Proposi- tion 5.1.2 that the set ˜D can be chosen as
˜
D = Dκ,mon+ :={S ∈ D+: 0 < S(t1)≤ S(t2)≤ κ ∀t1 ≤ t2}
if apriori information of the form
S†(t)≤ κ a. e. in I (6.25)
is available. Indeed, we have seen in Example 5.1.3 that the setD+
κ,mon is for every κ <∞ a
compactum in L2(I).
However, the application of Corollary 5.3.11 shows that the restriction of N to the set D+
κ :={S ∈ D+ : 0 < S(t)≤ κ a. e. in I}
has also a continuous inverse. In other words, the set ˜D can also be chosen as D+
κ. Combining
these considerations we can formulate the following result (cf. also [23, p. Theorem 4.2]). Corollary 6.2.3
Let{uδn}∞
n=1 be as sequence of noisy data and Sδn be a minimizer of (6.24) with uδ replaced
by uδn and either ˜D = D+
κ or ˜D = D+κ,mon. Let furthermore (6.25) be satisfied. Then the
convergence
uδn −−−→ uL2(I) † implies Sδn −−−→ SL2(I) †.
One can now ask which of the sets D+
κ or D+κ,mon should be preferred. On the one hand one
could argue that a†(t) > 0 a. e. and hence S†is strictly monotonically increasing. Therefore,
the additional apriori information which is incorporated in the choice ˜D = D+
κ,mon about the
monotonicity of S† is objective and can therefore not lead to a misspecification of the model.
On the contrary, it is preferable to include all available (and trustworthy) apriori information in the solution process as this might help to overcome the ill-conditioning effects. From the analytical point of view it seems therefore preferable to chose ˜D = D+
κ,mon.
On the other hand the following considerations show that the numerical realization of (6.24) with the choice ˜D = D+
κ is easier as with ˜D = D+κ,mon. Let us consider the following discreti-
sation of the problem. Let Vn denote the subspace of L2(I) spanned by the stepfunctions
that are constant on the intervals i−1n T,niT, i. e.
Vn:= ( n X i=1 g[i]χ (i−1 n T,n Ti ] (t) : g∈ Rn ) .
6.2. THE OUTER PROBLEM 99
Furthermore, let Vn
+ := Vn∩ L2+(R) denote the set of all nonnegative functions in Vn and
D+
κ := V+n∩ Dκ+ be the discretization of the set D+κ.
Defining the timepoints ti := niT −2n1 T we can discretized the operator N as
N: V+n→ Vn + [N(S)](t) := n X i=1 f (ti, S (ti)) χ (i−1 n T,n Ti ] (t) for S∈ VN + .
Besides, the range N (D+
κ) can be described as ( n X i=1 g[i]χ (i−1 n T,n Ti ] (t) : g∈ Rn, max P 0− Ke−rti ≤ g[i] ≤ f(ti, κ)∀i = 1, . . . , n ) .
Hence, for given ˜uδ ∈ Vn
+ the minimizer of
kN( ˜S)− uδk
L2(I) → min subject to ˜S ∈ D+κ (6.26)
can be computed as follows. 1. Compute ˜uδ∈ N (D+ κ) by ˜ uδ(t) := n X i=1 ˜ uδ[i]χ (i−1 n T,n Ti ] (t) with ˜uδ[i] := max P
0− Ke−rti, 0, min uδ(ti), f (ti, κ)
. 2. Compute ˜Sδ =Pn
i=1S˜δ[i]χ(i−1 n T,n Ti ]
(t) with ˜Sδ defined by
fti, ˜Sδ[i]
= ˜uδ[i] (i = 1, . . . , n) . (6.27)
We remark that in this step the continuity and the strict monotonicity of f (t, s) can be used. Indeed, for every i (6.27) can be solved by a simple bisection algorithm. In this way, the solution of the minimization problem (6.26) is decomposed into several independent subproblems, which reduces the complexity of the problem significantly.
As opposed to that the solution of kN( ˜S)− uδk
L2(I) → min subject to ˜S ∈ D+κ,mon (6.28)
with D+
κ,mon := V+n∩ D+κ,mon is more complicated. In this situation a decomposition of the
problem into independent subproblems is not possible as the restriction f[i] ≤ f[i + 1] has to be fulfilled. From the numerical point of view it might therefore be interesting to use descriptive regularization with the choice ˜D = D+
We conclude our considerations with respect to the setting (b) with the remark that an extension to the situation N :D(N) ⊂ Lp(I)→ Lq(I) with 1≤ p, q < ∞ is straightforward.
In fact, the construction of a sequence Sn such that (6.23) holds can be done analogously.
Furthermore, Corollary 5.3.11 which shows that the restriction of N to the set D+
κ :={S ∈ D+ : 0≤ S(t) ≤ κ a. e. in I}
has a continuous inverse is also valid for an Nemytskii operator acting between Lp(I) and
Lq(I) with 1 ≤ p, q < ∞. However, extensions of the result to situations with p = ∞ or
q =∞ seem not to be obvious. The C(I)-setting
In the rest of this section we confine our considerations to the setting (a). We start by formulating some consequences of Lemma 6.2.1 concerning properties of u†.
Remark 6.2.4 Combining Lemma 6.2.1 with Definitions (3.12) and (3.10) we get immedi- ately the following properties of u†(t) = f (t, S†(t)). We remark that properties 1 and 2 can
also be shown on more general arbitrage-free markets (cf.[21, p. 94ff]). 1. For all t > 0 we have
max(P0− Ke−rt, 0) < u†(t) < P0.
2. The function u†(t) is continuous and strictly monotonically increasing. Furthermore,
it holds
u†(0) = max(P0− K, 0) .
3. For a continuous volatility function, the term structure u†(t) is continuously differen-
tiable, where we have u†′(t) = ∂uBS ∂s (P0, K, r, t, S †(t))S†′(t) +∂uBS ∂t P0, K, r, t, S †(t) = P0φ(d†1(t)) a†(t) 2pS†(t) + Kre −rtΦ(d† 2(t)) > 0
with d†1 and d†2 defined by
d†1(t) := ln P0 K + rt + S†2(t) p S†(t) and d † 2(t) := d†1(t)− p S†(t) . Especially it holds u†′(t)≥ Kre−rtΦ(d†2(t)) . (6.29) In order to guarantee the existence of a solution Sδ ∈ D+
0 of the operator equation (6.16)
it is necessary that the noisy data uδ satisfy certain conditions. In fact, in [23, p. 1325] it
6.2. THE OUTER PROBLEM 101
with u†replaced by uδ there exists a uniquely defined function Sδ(t) (t∈ I), which satisfies
the equation
f t, Sδ(t)= uδ(t) t∈ I . (6.30) Furthermore, the solution Sδ is an element of D+
0 and Sδ is bounded by 0 < Sδ ≤ S t ∈ I
where S satisfies the equation
f (0, S) = max
t∈I u
δ(t) . (6.31)
Noting that the proof of this result does not make use of the strict monotonicity of uδ one
can omit this condition and formulate the following assumption guaranteeing the existence of a solution Sδ ∈ D+
0 of the operator equation (6.16).
Assumption 6.2.5 The data function uδ(t) is assumed to be continuous and to satisfy
uδ(0) = max(P0−K, 0) and max(P0−Ke−rt, 0) < uδ(t) < P0 for all t ∈ (0, T ] . (6.32)
Besides, the properties of the function f imply that the conditions (6.32) are also satisfied for any function u∈ N(D+
0 ). Together, these considerations show that Assumption 6.2.5 is
necessary and sufficient for the existence of a function Sδ satisfying (6.30). In other words, the range of N is equal to the set
C1 :={u ∈ C(I) : u(0) = max(P0− K, 0) and
max(P0− Ke−rt, 0) < u(t) < P0 ∀ t ∈ (0, T ]} .
We proceed by reviewing results of [23] concerning pointwise estimates of the absolute error |S†(t)− Sδ(t)|. Denoting by f
s(t, s) = ∂f∂s(t, s) = uBS(P0∂s,K,r,t,s)(t, s) the partial derivative of f
with respect to s and fixing t ∈ (0, T ] we can expand the function f into a Taylor series at the point S†(t). This gives
uδ(t) = f (t, Sδ(t)) = f (t, S†(t)) + fs(t, Sim(t)) Sδ(t)− S†(t)
, where Sim denotes a positive intermediate function such that
Sim(t)∈ [min(Sδ(t), S†(t)), max(Sδ(t), S†(t))] .
Because of (6.18) we can subtract u†(t) = f (t, S†(t)) on both sides and divide by f
s(t, Sim(t)).
Taking on both sides the absolute value gives |Sδ(t)− S†(t)| = 1
fs(t, Sim(t))|u
δ(t)− u†(t)| . (6.33)
Thus, for fixed S† and Sδ the function h(t) := 1
fs(t,Sim(t)) > 0 0 < t≤ T can be defined and
interpreted as error amplification factor. From the definition ofD+
0 follows lim t→0S †(t) = 0 and lim t→0S δ(t) = 0
and therefore also limt→0Sim(t) = 0. Now, for the situation P0 6= K the limit condition
(6.20) shows that limt→0h(t) =∞. Thus, in this situation for t ≈ 0 small errors in the data can lead to huge errors in the approximate solution.
These considerations show also that F−1 is not Lipschitz continuous, i. e. there cannot exist
any constant L, possibly depending on S0, such that an estimate of the form
kSδ− S0k
C(I)≤ Lkuδ− u0kC(I)
holds. If the operator N were linear, we could now conclude that the inverse operator N−1 :C1 ⊂ C(I) → D(N) ⊂ C(I)
is unbounded and thus not continuous. However, for the nonlinear problem the unbound- edness of the error amplification factor might either be a sign of ill-conditioning or of ill- posedness effects in a neighbourhood of t = 0.
Thus, the question whether the convergence
un −−→ uC(I) 0 implies N−1(un)−−→ NC(I) −1(u0)
(with elements un ∈ C1 n = 0, 1, 2, . . .) remained open. In [22] only pointwise convergence
of Sn := N−1(un) to S0 := N−1(u0) could be proven. Applying Theorem 5.3.6 we are now
able to answer this question. Theorem 6.2.6
Let P0, K > 0 and r ≥ 0. Then the operator N−1 :C1 ⊂ C(I) → C(I) is continuous, i. e.
kN(Sn)− N(S0)kC(I) → 0 implies kSn− S0kC(I) → 0.
Together with the injectivity of N we conclude from Theorem 6.2.6 that the inverse problem N(Sδ) = uδ (Sδ ∈ D(N), uδ ∈ C
1) (6.34)
is well-posed.
Nevertheless, we have seen that for P0 6= K there occur ill-conditioning effects in the sense
that small noise in the data uδ can be strongly amplified. In terms of convergence one
can conclude that for a sequence of noisy data uδn ∈ C
1 converging to the true data u†
the corresponding solutions Sδn = N−1(uδn) do also converge to S† = N−1(u†) but the
convergence can be very slow. One could therefore speak of delayed convergence. In the following subsection we will find a conjecture about the convergence rate.